Statistical physics is the natural framework to model complex networks. In the last twenty years, it has brought novel physical insights on a variety of emergent phenomena, such as self-organisation, scale invariance, mixed distributions and ensemble non-equivalence, which cannot be deduced from the behaviour of the individual constituents. At the same time, thanks to its deep connection with information theory, statistical physics and the principle of maximum entropy have led to the definition of null models reproducing some features of empirical networks, but otherwise as random as possible. We review here the statistical physics approach for complex networks and the null models for the various physical problems, focusing in particular on the analytic frameworks reproducing the local features of the network. We show how these models have been used to detect statistically significant and predictive structural patterns in real-world networks, as well as to reconstruct the network structure in case of incomplete information. We further survey the statistical physics frameworks that reproduce more complex, semi-local network features using Markov chain Monte Carlo sampling, and the models of generalised network structures such as multiplex networks, interacting networks and simplicial complexes.

Comments: To appear on Nature Reviews Physics. The revised accepted version will be posted 6 months after publication

A country's mix of products predicts its subsequent pattern of diversification and economic growth. But does this product mix also predict income inequality? Here we combine methods from econometrics, network science, and economic complexity to show that countries exporting complex products (as measured by the Economic Complexity Index) have lower levels of income inequality than countries exporting simpler products. Using multivariate regression analysis, we show that economic complexity is a significant and negative predictor of income inequality and that this relationship is robust to controlling for aggregate measures of income, institutions, export concentration, and human capital. Moreover, we introduce a measure that associates a product to a level of income inequality equal to the average GINI of the countries exporting that product (weighted by the share the product represents in that country's export basket). We use this measure together with the network of related products (or product space) to illustrate how the development of new products is associated with changes in income inequality. These findings show that economic complexity captures information about an economy's level of development that is relevant to the ways an economy generates and distributes its income. Moreover, these findings suggest that a country's productive structure may limit its range of income inequality. Finally, we make our results available through an online resource that allows for its users to visualize the structural transformation of over 150 countries and their associated changes in income inequality between 1963 and 2008.